NAG Library Routine Document

f02jcf  (real_gen_quad)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{scal}$ – IntegerInput

On entry: determines the form of scaling to be performed on $A$, $B$ and $C$.

$\mathbf{scal}=0$

No scaling.

$\mathbf{scal}=1$ (the recommended value)

Fan, Lin and Van Dooren scaling if $\frac{‖B‖}{\sqrt{‖A‖\times ‖C‖}}<10$ and no scaling otherwise where $‖Z‖$ is the Frobenius norm of $Z$.

$\mathbf{scal}=2$

Fan, Lin and Van Dooren scaling.

$\mathbf{scal}=3$

Tropical scaling with largest root.

$\mathbf{scal}=4$

Tropical scaling with smallest root.

Constraint: $\mathbf{scal}=0$, $1$, $2$, $3$ or $4$.

2:     $\mathbf{jobvl}$ – Character(1)Input

On entry: if $\mathbf{jobvl}=\text{'N'}$, do not compute left eigenvectors.

If $\mathbf{jobvl}=\text{'V'}$, compute the left eigenvectors.

If $\mathbf{sense}=1$, $2$, $4$, $5$, $6$ or $7$, jobvl must be set to 'V'.

Constraint: $\mathbf{jobvl}=\text{'N'}$ or $\text{'V'}$.

3:     $\mathbf{jobvr}$ – Character(1)Input

On entry: if $\mathbf{jobvr}=\text{'N'}$, do not compute right eigenvectors.

If $\mathbf{jobvr}=\text{'V'}$, compute the right eigenvectors.

If $\mathbf{sense}=1$, $3$, $4$, $5$, $6$ or $7$, jobvr must be set to 'V'.

Constraint: $\mathbf{jobvr}=\text{'N'}$ or $\text{'V'}$.

4:     $\mathbf{sense}$ – IntegerInput

On entry: determines whether, or not, condition numbers and backward errors are computed.

$\mathbf{sense}=0$

Do not compute condition numbers, or backward errors.

$\mathbf{sense}=1$

Just compute condition numbers for the eigenvalues.

$\mathbf{sense}=2$

Just compute backward errors for the left eigenpairs.

$\mathbf{sense}=3$

Just compute backward errors for the right eigenpairs.

$\mathbf{sense}=4$

Compute backward errors for the left and right eigenpairs.

$\mathbf{sense}=5$

Compute condition numbers for the eigenvalues and backward errors for the left eigenpairs.

$\mathbf{sense}=6$

Compute condition numbers for the eigenvalues and backward errors for the right eigenpairs.

$\mathbf{sense}=7$

Compute condition numbers for the eigenvalues and backward errors for the left and right eigenpairs.

Constraint: $\mathbf{sense}=0$, $1$, $2$, $3$, $4$, $5$, $6$ or $7$.

5:     $\mathbf{tol}$ – Real (Kind=nag_wp)Input

On entry: tol is used as the tolerance for making decisions on rank in the deflation procedure. If tol is zero on entry then $n\times \mathit{machine precision}$ is used in place of tol, where machine precision is as returned by routine x02ajf. A diagonal element of a triangular matrix, $R$, is regarded as zero if $\left|r_{jj}\right|\le \mathbf{tol}\times \mathrm{size}\left(X\right)$, or $n\times \mathit{machine precision}\times \mathrm{size}\left(X\right)$ when tol is zero, where $\mathrm{size}\left(X\right)$ is based on the size of the absolute values of the elements of the matrix $X$ containing the matrix $R$. See Hammarling et al. (2013) for the motivation. If tol is $-1.0$ on entry then no deflation is attempted. The recommended value for tol is zero.

6:     $\mathbf{n}$ – IntegerInput

On entry: the order of the matrices $A$, $B$ and $C$.

Constraint: $\mathbf{n}\ge 0$.

7:     $\mathbf{a}\left(\mathbf{lda},*\right)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array a must be at least $\mathbf{n}$.

On entry: the $n$ by $n$ matrix $A$.

On exit: a is used as internal workspace, but if $\mathbf{jobvl}=\text{'V'}$ or $\mathbf{jobvr}=\text{'V'}$, a is restored on exit.

8:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f02jcf is called.

Constraint: $\mathbf{lda}\ge \mathbf{n}$.

9:     $\mathbf{b}\left(\mathbf{ldb},*\right)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array b must be at least $\mathbf{n}$.

On entry: the $n$ by $n$ matrix $B$.

On exit: b is used as internal workspace, but is restored on exit.

10:   $\mathbf{ldb}$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f02jcf is called.

Constraint: $\mathbf{ldb}\ge \mathbf{n}$.

11:   $\mathbf{c}\left(\mathbf{ldc},*\right)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array c must be at least $\mathbf{n}$.

On entry: the $n$ by $n$ matrix $C$.

On exit: c is used as internal workspace, but if $\mathbf{jobvl}=\text{'V'}$ or $\mathbf{jobvr}=\text{'V'}$, c is restored on exit.

12:   $\mathbf{ldc}$ – IntegerInput

On entry: the first dimension of the array c as declared in the (sub)program from which f02jcf is called.

Constraint: $\mathbf{ldc}\ge \mathbf{n}$.

13:   $\mathbf{alphar}\left(2\times \mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{alphar}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,2n$, contains the real part of ${\alpha }_j$ for the $j$th eigenvalue pair $\left({\alpha }_j,{\beta }_j\right)$ of the quadratic eigenvalue problem.

14:   $\mathbf{alphai}\left(2\times \mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{alphai}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,2n$, contains the imaginary part of ${\alpha }_j$ for the $j$th eigenvalue pair $\left({\alpha }_j,{\beta }_j\right)$ of the quadratic eigenvalue problem. If $\mathbf{alphai}\left(j\right)$ is zero then the $j$th eigenvalue is real; if $\mathbf{alphai}\left(j\right)$ is positive then the $j$th and $\left(j+1\right)$th eigenvalues are a complex conjugate pair, with $\mathbf{alphai}\left(j+1\right)$ negative.

15:   $\mathbf{beta}\left(2\times \mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{beta}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,2n$, contains the second part of the $j$th eigenvalue pair $\left({\alpha }_j,{\beta }_j\right)$ of the quadratic eigenvalue problem, with ${\beta }_j\ge 0$. Infinite eigenvalues have ${\beta }_j$ set to zero.

16:   $\mathbf{vl}\left(\mathbf{ldvl},*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the second dimension of the array vl must be at least $2\times \mathbf{n}$ if $\mathbf{jobvl}=\text{'V'}$.

On exit: if $\mathbf{jobvl}=\text{'V'}$, the left eigenvectors $y_j$ are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues. If the $j$th eigenvalue is real, then $y_j=\mathbf{vl}\left(:,j\right)$, the $j$th column of vl. If the $j$th and $\left(j+1\right)$th eigenvalues form a complex conjugate pair, then $y_j=\mathbf{vl}\left(:,j\right)+i\times \mathbf{vl}\left(:,j+1\right)$ and $y_{j+1}=\mathbf{vl}\left(:,j\right)-i\times \mathbf{vl}\left(:,j+1\right)$. Each eigenvector will be normalized with length unity and with the element of largest modulus real and positive.

If $\mathbf{jobvl}=\text{'N'}$, vl is not referenced.

17:   $\mathbf{ldvl}$ – IntegerInput

On entry: the first dimension of the array vl as declared in the (sub)program from which f02jcf is called.

Constraint: $\mathbf{ldvl}\ge \mathbf{n}$.

18:   $\mathbf{vr}\left(\mathbf{ldvr},*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the second dimension of the array vr must be at least $2\times \mathbf{n}$ if $\mathbf{jobvr}=\text{'V'}$.

On exit: if $\mathbf{jobvr}=\text{'V'}$, the right eigenvectors $x_j$ are stored one after another in the columns of vr, in the same order as the corresponding eigenvalues. If the $j$th eigenvalue is real, then $x_j=\mathbf{vr}\left(:,j\right)$, the $j$th column of vr. If the $j$th and $\left(j+1\right)$th eigenvalues form a complex conjugate pair, then $x_j=\mathbf{vr}\left(:,j\right)+i\times \mathbf{vr}\left(:,j+1\right)$ and $x_{j+1}=\mathbf{vr}\left(:,j\right)-i\times \mathbf{vr}\left(:,j+1\right)$. Each eigenvector will be normalized with length unity and with the element of largest modulus real and positive.

If $\mathbf{jobvr}=\text{'N'}$, vr is not referenced.

19:   $\mathbf{ldvr}$ – IntegerInput

On entry: the first dimension of the array vr as declared in the (sub)program from which f02jcf is called.

Constraint: $\mathbf{ldvr}\ge \mathbf{n}$.

20:   $\mathbf{s}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the array s must be at least $2\times \mathbf{n}$ if $\mathbf{sense}=1$, $5$, $6$ or $7$.

Also: computing the condition numbers of the eigenvalues requires that both the left and right eigenvectors be computed.

On exit: if $\mathbf{sense}=1$, $5$, $6$ or $7$, $\mathbf{s}\left(j\right)$ contains the condition number estimate for the $j$th eigenvalue (large condition numbers imply that the problem is near one with multiple eigenvalues). Infinite condition numbers are returned as the largest model real number (x02alf).

If $\mathbf{sense}=0$, $2$, $3$ or $4$, s is not referenced.

21:   $\mathbf{bevl}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the array bevl must be at least $2\times \mathbf{n}$ if $\mathbf{sense}=2$, $4$, $5$ or $7$.

On exit: if $\mathbf{sense}=2$, $4$, $5$ or $7$, $\mathbf{bevl}\left(j\right)$ contains the backward error estimate for the computed left eigenpair $\left({\lambda }_j,y_j\right)$.

If $\mathbf{sense}=0$, $1$, $3$ or $6$, bevl is not referenced.

22:   $\mathbf{bevr}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the array bevr must be at least $2\times \mathbf{n}$ if $\mathbf{sense}=3$, $4$, $6$ or $7$.

On exit: if $\mathbf{sense}=3$, $4$, $6$ or $7$, $\mathbf{bevr}\left(j\right)$ contains the backward error estimate for the computed right eigenpair $\left({\lambda }_j,x_j\right)$.

If $\mathbf{sense}=0$, $1$, $2$ or $5$, bevr is not referenced.

23:   $\mathbf{iwarn}$ – IntegerOutput

On exit: iwarn will be positive if there are warnings, otherwise iwarn will be $0$.

If $\mathbf{ifail}=\mathbf{0}$ then:

• if $\mathbf{iwarn}=1$ then one, or both, of the matrices $A$ and $C$ is zero. In this case no scaling is performed, even if $\mathbf{scal}>0$;
• if $\mathbf{iwarn}=2$ then the matrices $A$ and $C$ are singular, or nearly singular, so the problem is potentially ill-posed;
• if $\mathbf{iwarn}=3$ then both the conditions for $\mathbf{iwarn}=1$ and $\mathbf{iwarn}=2$ above, apply. If $\mathbf{iwarn}=4$, $‖\mathbf{b}‖\ge 10\sqrt{‖A‖·‖C‖}$ and backward stability cannot be guaranteed.

If $\mathbf{ifail}=\mathbf{2}$, iwarn returns the value of info from f08xaf (dgges).

If $\mathbf{ifail}=\mathbf{3}$, iwarn returns the value of info from f08waf (dggev).

24:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

The quadratic matrix polynomial is nonregular (singular).

$\mathbf{ifail}=2$

The $QZ$ iteration failed in f08xaf (dgges).
iwarn returns the value of info returned by f08xaf (dgges). This failure is unlikely to happen, but if it does, please contact NAG.

$\mathbf{ifail}=3$

The $QZ$ iteration failed in f08waf (dggev).
iwarn returns the value of info returned by f08waf (dggev). This failure is unlikely to happen, but if it does, please contact NAG.

$\mathbf{ifail}=-1$

On entry, $\mathbf{scal}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{scal}=0$, $1$, $2$, $3$ or $4$.

$\mathbf{ifail}=-2$

On entry, $\mathbf{jobvl}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{jobvl}=\text{'N'}$ or $\text{'V'}$.

On entry, $\mathbf{sense}=〈\mathit{\text{value}}〉$ and $\mathbf{jobvl}=〈\mathit{\text{value}}〉$.
Constraint: when $\mathbf{jobvl}=\text{'N'}$, $\mathbf{sense}=0$ or $3$,
when $\mathbf{jobvl}=\text{'V'}$, $\mathbf{sense}=1$, $2$, $4$, $5$, $6$ or $7$.

$\mathbf{ifail}=-3$

On entry, $\mathbf{jobvr}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{jobvr}=\text{'N'}$ or $\text{'V'}$.

On entry, $\mathbf{sense}=〈\mathit{\text{value}}〉$ and $\mathbf{jobvr}=〈\mathit{\text{value}}〉$.
Constraint: when $\mathbf{jobvr}=\text{'N'}$, $\mathbf{sense}=0$ or $2$,
when $\mathbf{jobvr}=\text{'V'}$, $\mathbf{sense}=1$, $3$, $4$, $5$, $6$ or $7$.

$\mathbf{ifail}=-4$

On entry, $\mathbf{sense}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{sense}=0$, $1$, $2$, $3$, $4$, $5$, $6$ or $7$.

$\mathbf{ifail}=-6$

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 0$.

$\mathbf{ifail}=-8$

On entry, $\mathbf{lda}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lda}\ge \mathbf{n}$.

$\mathbf{ifail}=-10$

On entry, $\mathbf{ldb}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ldb}\ge \mathbf{n}$.

$\mathbf{ifail}=-12$

On entry, $\mathbf{ldc}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ldc}\ge \mathbf{n}$.

$\mathbf{ifail}=-17$

On entry, $\mathbf{ldvl}=〈\mathit{\text{value}}〉$, $\mathbf{n}=〈\mathit{\text{value}}〉$ and $\mathbf{jobvl}=〈\mathit{\text{value}}〉$.
Constraint: when $\mathbf{jobvl}=\text{'V'}$, $\mathbf{ldvl}\ge \mathbf{n}$.

$\mathbf{ifail}=-19$

On entry, $\mathbf{ldvr}=〈\mathit{\text{value}}〉$, $\mathbf{n}=〈\mathit{\text{value}}〉$ and $\mathbf{jobvr}=〈\mathit{\text{value}}〉$.
Constraint: when $\mathbf{jobvr}=\text{'V'}$, $\mathbf{ldvr}\ge \mathbf{n}$.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f02jcfe.f90)

10.2

Program Data

Program Data (f02jcfe.d)

10.3

Program Results

Program Results (f02jcfe.r)